On approximation of Ginzburg-Landau minimizers by $\mathbb S^1$-valued maps in domains with vanishingly small holes

Leonid Berlyand; Dmitry Golovaty; Oleksandr Iaroshenko; Volodymyr Rybalko

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arXiv · 1701.01534

On approximation of Ginzburg-Landau minimizers by $\mathbb S^1$-valued maps in domains with vanishingly small holes

Leonid Berlyand·Dmitry Golovaty·Oleksandr Iaroshenko·Volodymyr Rybalko

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Abstract

We consider a two-dimensional Ginzburg-Landau problem on an arbitrary domain with a finite number of vanishingly small circular holes. A special choice of scaling relation between the material and geometric parameters (Ginzburg-Landau parameter vs hole radius) is motivated by a recently dsicovered phenomenon of vortex phase separation in superconducting composites. We show that, for each hole, the degrees of minimizers of the Ginzburg-Landau problems in the classes of $\mathbb S^1$-valued and $\mathbb C$-valued maps, respectively, are the same. The presence of two parameters that are widely separated on a logarithmic scale constitutes the principal difficulty of the analysis that is based on energy decomposition techniques.

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Leonid Berlyand, Dmitry Golovaty, Oleksandr Iaroshenko, Volodymyr Rybalko. 2017-01-06. On approximation of Ginzburg-Landau minimizers by $\mathbb S^1$-valued maps in domains with vanishingly small holes. https://arxiv.org/abs/1701.01534

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